He showed that belief by building schools and inviting scholars to his court.
This is the best way to combine the two sentences. The parallel structure is correct with both phrases beginning with the -ing form of the verb followed by the direct object. And is the correct conjunction for the sentence because Alfred the Great both built schools and invited scholars. Using while to join the sentences makes it seem that the one event relies on the other. But and yet are also not accurate conjunctions because they indicate a incorrect relationship between the ideas.
